Consider the system where r² = x² + y². J = -y+x(r4-3r²+1), \ ý = x + y(r²¹ - 3r²+1), (a) Use the Poincaré-Bendixson Theorem to show that there is a periodic orbot in the annular region A₁ = {(x, y)|1 < √√x² + y² <2}. (b) Show that the origin is an unstable focus for this system and use the Poincaré- Bendixson Theorem to show that there is periodic orbit in the annular region A₂ = {(x, y)le < √² + y² < 1}, where 0 < € <<< 1.
Consider the system where r² = x² + y². J = -y+x(r4-3r²+1), \ ý = x + y(r²¹ - 3r²+1), (a) Use the Poincaré-Bendixson Theorem to show that there is a periodic orbot in the annular region A₁ = {(x, y)|1 < √√x² + y² <2}. (b) Show that the origin is an unstable focus for this system and use the Poincaré- Bendixson Theorem to show that there is periodic orbit in the annular region A₂ = {(x, y)le < √² + y² < 1}, where 0 < € <<< 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider the system
where 2 = x² + y².
y
ý
=
=
y + x(r² - 3r²+1),
x+y(r² = 3r²+1),
(a) Use the Poincaré-Bendixson Theorem to show that there is a periodic orbot in the
annular region A₁ = {(x, y)]1 < √√√x² + y² <2}.
(b) Show that the origin is an unstable focus for this system and use the Poincaré-
Bendixson Theorem to show that there is periodic orbit in the annular region
A₂ = {(x, y)<√√x² + y² <1}, where 0 < € <<< 1.
(Hint: You may need the fact that the only equilibrium of system is at the origin.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bfc7ddd-9456-4dbb-af62-a90660d5f5ef%2F72d45a56-3848-4231-a219-7ca13ba5eec5%2Fjllm4ji_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the system
where 2 = x² + y².
y
ý
=
=
y + x(r² - 3r²+1),
x+y(r² = 3r²+1),
(a) Use the Poincaré-Bendixson Theorem to show that there is a periodic orbot in the
annular region A₁ = {(x, y)]1 < √√√x² + y² <2}.
(b) Show that the origin is an unstable focus for this system and use the Poincaré-
Bendixson Theorem to show that there is periodic orbit in the annular region
A₂ = {(x, y)<√√x² + y² <1}, where 0 < € <<< 1.
(Hint: You may need the fact that the only equilibrium of system is at the origin.)
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